Real functions and graphs

Sometimes the best way to understand a set of data is to sketch a simple graph. This...

Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques.

By the end of this section you should be able to:

understand the definition of a real function;

use the notation for intervals of the real line;

recognise and use the graphs of the basic functions described in the audio section;

understand the effect on a graph of translations, scalings, rotations and reflections;

understand how the shape of a graph of a function features properties of the function such as increasing, decreasing, even and odd.

determine the x-intercepts and y-intercept of a given function f;

determine the intervals on which a given function f is positive or negative;

determine the intervals on which a given function f is increasing or decreasing, and any points at which f has a local maximum or local minimum;

describe the asymptotic behaviour (if any) of a given function f;

sketch the graph of a given function.

sketch the graph of a combination of two functions, one of which is a trigonometric function;

sketch the graph of a hybrid function, whose rule is defined by different formulas on different parts of its domain.

define the hyperbolic functions cosh x, sinh x and tanh x, and be familiar with their properties;

sketch the graphs of cosh x, sinh x and tanh x, and their reciprocals.

plot a curve that is specified by a parametric representation;

obtain the equation of a curve that is specified by a parametric representation, in simple cases;

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Real functions and graphs

Introduction

Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply ‘joining up the dots’ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of this unit is to review the various techniques for sketching graphs that you may have met in your previous studies, and to extend these methods.

This unit is an adapted extract from the Open University course Pure mathematics
(M208) [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]

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Originally published: Tuesday, 28th June 2011

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